Abstract
In spite of being limited to the solution of only certain types of PDEs, the method of separation of variables provides often an avenue of approach to large classes of problems and affords important physical insights. An example of this kind is provided by the analysis of vibration problems in Engineering. The separation of variables in this case results in the resolution of a hyperbolic problem into a series of elliptic problems. The same idea will be applied in another chapter to resolve a parabolic equation in a similar manner. One of the main by-products of the method is the appearance of a usually discrete spectrum of natural properties acting as a natural signature of the system. This feature is particularly manifest in diverse applications, from musical acoustics to Quantum Mechanics.
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- 1.
Historically, in fact, the method was discovered by Jean Baptiste Joseph Fourier (1768–1830) in his celebrated book Théorie Analytique de la Chaleur. Thus, its very first application lies within the realm of parabolic equations.
- 2.
The linearity of the stiffness properties, which results in the linearity of the equations of motion, is either an inherent property of the system or, alternatively, the consequence of assuming that the displacements of the system are very small in some precise sense.
- 3.
It is best here to think of the case of small rigid-body motions only.
- 4.
An excellent reference is [2].
- 5.
The treatment for the general case is identical, except for the fact that normal modes of the form \(A_i +B_it\) must be included.
- 6.
In Eq. (9.18) the summation convention does not apply.
- 7.
We content ourselves with pointing out these similarities. In fact these similarities run even deeper, particularly when we regard the underlying differential equations as linear operators on an infinite-dimensional vector space of functions, just as a matrix is a linear operator on a finite-dimensional space of vectors.
- 8.
A short treatment can be found in [1], p. 291.
- 9.
See Exercise 9.6.
- 10.
In particular, the Duhamel integral will have to be expressed differently as compared to the uniform case.
- 11.
For a proof, see [1], p. 359.
- 12.
As a matter of historical interest, the Neumann boundary condition is named after the German mathematician Carl Neumann (1832–1925), not to be confused with the Hungarian-American mathematician John von Neumann (1903–1957).
References
Courant R, Hilbert D (1962) Methods of mathematical physics, vol I. Interscience, Wiley, New York
Hildebrand FB (1965) Methods of applied mathematics. Prentice Hall, Englewood Cliffs (Reprinted by Dover (1992))
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Epstein, M. (2017). Standing Waves and Separation of Variables. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_9
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DOI: https://doi.org/10.1007/978-3-319-55212-5_9
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